Finite Group Databasebeta

← #214452

Subgroups — #214452

Order 512.

999 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35111C211yesmaximal, characteristic
2#21822C211yesmaximal, characteristic, complemented
3#13942C211yesmaximal, characteristic, complemented
4#9156C211yesmaximal, characteristic
5#9152C211yesmaximal, characteristic, complemented
6#9148C211yesmaximal, characteristic
7#9144C211yesmaximal, characteristic, complemented
8#9118C212yesmaximal
9#9116C212yesmaximal, complemented
10#8714C212yesmaximal, complemented
11#8330C212yesmaximal
12C2×#529C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#432C2212yes—
15#2855C2211yescharacteristic
16#2855C2211yescharacteristic
17#2854C2211yescharacteristic
18#2854C2211yescharacteristic
19#2623—21no—
20#2614—21no—
21#2604—22no—
22#2599—22no—
23#2592C412yescomplemented
24#2584C412yescomplemented
25#2581C412yescomplemented
26#2580C412yescomplemented
27#2322—21no—
28#2322—21no—
29C2×#391—21no—
30#2283C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1985—22no—
34#1985—22no—
35#1817C2212yes—
36#1817C2211yescharacteristic
37#1817C2211yescharacteristic
38C2×#328C2212yes—
39C2×#328C2212yes—
40C2×#327C2212yes—
41C2×#327C2212yes—
42#1441—21no—
43#1441—21no—
44C4⋊#95—21no—
45#1434C2212yes—
46#1434C2212yes—
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51C23×Q8D411yescharacteristic
52#529—42no—
53#529—42no—
54C2×#126—21no—
55C2×#125C2×C411yescharacteristic
56C2×#124C2×C411yescharacteristic
57C2×#123—21no—
58C2×#123—21no—
59C2×#123—21no—
60C2×C22⋊Q8C2311yescharacteristic
61C2×#121C2311yescharacteristic
62C2×C4×Q8—42no—
63C2×C4×Q8—42no—
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8C2×C412yes—
66C2×C4×Q8C2×C412yes—
67C2×C4×Q8D412yes—
68C2×C4×Q8D412yes—
69C2×C4×D4—21no—
70C2×C4×D4C2×C411yescharacteristic
71C2×#117—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81#432—42no—
82#432—42no—
83C2×C4⋊C8—22no—
84C2×C22⋊C8C2312yes—
85C2×C22⋊C8C2312yes—
86C2×#97—22no—
87#400—22no—
88#399—22no—
89#398—22no—
90#397—22no—
91#395—22no—
92#395—22no—
93#393—22no—
94#393—22no—
95#391—44no—
96#391C2×C412yes—
97#391C2×C412yes—
98#390C2×C412yes—
99#390C2×C412yes—
100#389—22no—
101#389—22no—
102#389C2×C412yes—
103#389C2×C412yes—
104#389C2×C412yes—
105#389C2×C412yes—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#386—21no—
115#386—21no—
116#385—21no—
117#385—21no—
118#385—22no—
119#385—22no—
120#383—21no—
121#382—42no—
122#382C2×C412yes—
123#381—21no—
124#381—21no—
125#381—21no—
126#380C2311yescharacteristic
127#380C2311yescharacteristic
128C4×C4⋊C4—42no—
129C4×C4⋊C4C2×C412yes—
130C4×C22⋊C4—21no—
131#376—21no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—21no—
137C2×#95C2312yes—
138C2×#95C2312yes—
139#336—22no—
140#336—22no—
141#336—22no—
142#336—22no—
143#328—42no—
144#328—42no—
145#327—42no—
146#327—42no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—41no—
150C22×Q8—41no—
151C22×Q8—41no—
152C22×Q8—41no—
153C22×Q8C22⋊C412yes—
154C22×Q8C22⋊C412yes—
155C22×Q8C22×C411yescharacteristic
156C22×D4—21no—
157C22×D4C22×C411yescharacteristic
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2411yescharacteristic
168C22×C8—22no—
169C22×C8—22no—
170#126—44no—
171#125—42no—
172#124—42no—
173#123—42no—
174#123—42no—
175#123—42no—
176#123—42no—
177#123—42no—
178#123—42no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181#121—22no—
182#121—22no—
183C4×Q8—44no—
184C4×Q8—44no—
185C4×Q8—42no—
186C4×Q8—42no—
187C4×Q8—42no—
188C4×Q8—42no—
189C4×Q8—42no—
190C4×D4—42no—
191C4×D4—42no—
192C4×D4—42no—
193#117—44no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42C22×C411yescharacteristic
246C2×C42C22×C411yescharacteristic
247C2×C42C22×C412yes—
248C2×C42C22×C412yes—
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C4⋊C8—44no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258#97—44no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—21no—
271#95—21no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—44no—
277#95—44no—
278#95—42no—
279#95—44no—
280#95—44no—
281C24—21no—
282C24C22×D411yescharacteristic
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—22no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—44no—
291C2×Q8—44no—
292C2×Q8—44no—
293C2×Q8—44no—
294C2×Q8—42no—
295C2×D4—22no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—82no—
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C412yes—
358C22×C4AGL(1,ℤ/8)12yes—
359C22×C4#13712yes—
360C22×C4C23×C411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4H(5,𝔽2)12yes—
363C22×C4H(5,𝔽2)12yes—
364C2×C8—44no—
365C2×C8—44no—
366C2×C8—42no—
367C2×C8—42no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—48no—
383C4⋊C4—42no—
384C4⋊C4—44no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4—84no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—44no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—42no—
399C22⋊C4—44no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C42—22no—
403C42—24no—
404C42—24no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—42no—
413C42—82no—
414C42—84no—
415C23—22no—
416C23—22no—
417C23—41no—
418C23—42no—
419C23—41no—
420C23—41no—
421C23—42no—
422C23C22×C22⋊C411yescharacteristic
423C23#53411yescharacteristic
424C23#53411yescharacteristic
425C23C2×AGL(1,ℤ/8)11yescharacteristic
426C23C2×#13711yescharacteristic
427C23#57511yescharacteristic
428C23#57511yescharacteristic
429Q8—48no—
430Q8—48no—
431Q8—82no—
432D4—82no—
433D4—81no—
434D4—81no—
435D4—81no—
436D4—81no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—84no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4#41714yes—
509C2×C4#51812yes—
510C2×C4#53412yes—
511C8—82no—
512C8—82no—
513C22—21no—
514C22—21no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—21no—
520C22—21no—
521C22—81no—
522C22—81no—
523C22—82no—
524C22—82no—
525C22—81no—
526C22#230811yescharacteristic
527C22#288412yes—
528C22C2×#41711yescharacteristic
529C22#289611yescharacteristic
530C22#290112yes—
531C22#319411yescharacteristic
532C22#319411yescharacteristic
533C22#319611yescharacteristic
534C22#319611yescharacteristic
535C4—24no—
536C4—22no—
537C4—22no—
538C4—41no—
539C4—41no—
540C4—42no—
541C4—42no—
542C4—41no—
543C4—42no—
544C4—41no—
545C4—84no—
546C4—81no—
547C4—81no—
548C4—82no—
549C4—82no—
550C4—81no—
551C4—162no—
552C2—21no—
553C2—21no—
554C2—21no—
555C2—21no—
556C2—161no—
557C2#1762412yes—
558C2#1762812yes—
559C2#2175111yescharacteristic
560C2#2176911yescharacteristic
561C2#2177511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21822C2semidirect
3. #13942C2semidirect
5. #9152C2semidirect
7. #9144C2semidirect
9. #9116C2semidirect
10. #8714C2semidirect
23. #2592C4semidirect
24. #2584C4semidirect
25. #2581C4semidirect
26. #2580C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#351112
#218222
#139422
#91562
#91522
#91482
#91442
#91182
#91162
#87142
#83302